Ideal Hyperbolic Polyhedra and Discrete Uniformization

Abstract

We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmüller spaces \(\widetilde{\mathscr {T}}_{g,n}\) of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over \(\mathscr {T}_{g,n}\), and invariant under the action of the mapping class group.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

References

  1. 1.

    Akiyoshi, H.: Finiteness of polyhedral decompositions of cusped hyperbolic manifolds obtained by the Epstein–Penner’s method. Proc. Am. Math. Soc. 129(8), 2431–2439 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005)

    Google Scholar 

  3. 3.

    Aurenhammer, F., Klein, R., Lee, D.-T.: Voronoi Diagrams and Delaunay Triangulations. World Scientific, Hackensack (2013)

    MATH  Book  Google Scholar 

  4. 4.

    Bao, X., Bonahon, F.: Hyperideal polyhedra in hyperbolic 3-space. Bull. Soc. Math. France 130(3), 457–491 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Bobenko, A.I., Izmestiev, I.: Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58(2), 447–505 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bobenko, A.I., Dimitrov, N., Sechelmann, S.: Discrete uniformization of polyhedral surfaces with non-positive curvature and branched covers over the sphere via hyper-ideal circle patterns. Discrete Comput. Geom. 57(2), 431–469 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Bobenko, A.I., Pinkall, U., Springborn, B.A.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19(4), 2155–2215 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bobenko, A.I., Sechelmann, S., Springborn, B.: Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization. In: Bobenko, A.I. (ed.) Advances in Discrete Differential Geometry, pp. 1–56. Springer, Berlin (2016)

    Google Scholar 

  9. 9.

    Bobenko, A.I., Springborn, B.A.: A discrete Laplace–Beltrami operator for simplicial surfaces. Discrete Comput. Geom. 38(4), 740–756 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bowers, J.C., Bowers, P.L., Pratt, K.: Rigidity of circle polyhedra in the 2-sphere and of hyperideal polyhedra in hyperbolic 3-space. Trans. Am. Math. Soc. 371(6), 4215–4249 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Duffin, R.J.: Distributed and lumped networks. J. Math. Mech. 8, 793–826 (1959)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge Monographs on Applied and Computational Mathematics, vol. 7. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  13. 13.

    Epstein, D.B.A., Penner, R.C.: Euclidean decompositions of noncompact hyperbolic manifolds. J. Differ. Geom. 27(1), 67–80 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Fillastre, F.: Polyhedral hyperbolic metrics on surfaces. Geom. Dedicata 134, 177–196 (2008). Erratum 138, 193–194 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Fillastre, F., Izmestiev, I.: Hyperbolic cusps with convex polyhedral boundary. Geom. Topol. 13(1), 457–492 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Fortune, S.: Numerical stability of algorithms for 2D Delaunay triangulations. Int. J. Comput. Geom. Appl. 5(1–2), 193–213 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Gu, X., Guo, R., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces II. J. Differ. Geom. 109(3), 431–466 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Gu, X.D., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces. J. Differ. Geom. 109(2), 223–256 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Indermitte, C., Liebling, T.M., Troyanov, M., Clémençon, H.: Voronoi diagrams on piecewise flat surfaces and an application to biological growth. Theoret. Comput. Sci. 263(1–2), 263–274 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Izmestiev, I.: A variational proof of Alexandrov’s convex cap theorem. Discrete Comput. Geom. 40(4), 561–585 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Joswig, M., Löwe, R., Springborn, B.: Secondary fans and secondary polyhedra of punctured Riemann surfaces. Exp. Math. (2019). https://doi.org/10.1080/10586458.2018.1477078

  22. 22.

    Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Masur, H., Smillie, J.: Hausdorff dimension of sets of nonergodic measured foliations. Ann. Math. 134(3), 455–543 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Milnor, J.: Hyperbolic geometry: the first 150 years. Bull. Am. Math. Soc. (N.S.) 6(1), 9–24 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Moroianu, S., Schlenker, J.-M.: Quasi-Fuchsian manifolds with particles. J. Differ. Geom. 83(1), 75–129 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Penner, R.C.: The decorated Teichmüller space of punctured surfaces. Commun. Math. Phys. 113(2), 299–339 (1987)

    MATH  Article  Google Scholar 

  27. 27.

    Penner, R.C.: Decorated Teichmüller Theory. QGM Master Class Series. European Mathematical Society (EMS), Zürich (2012)

    Book  Google Scholar 

  28. 28.

    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Prosanov, R.: Ideal polyhedral surfaces in Fuchsian manifolds. arXiv:1804.05893 (2018)

  30. 30.

    Rippa, S.: Minimal roughness property of the Delaunay triangulation. Comput. Aided Geom. Des. 7(6), 489–497 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Rivin, I.: Intrinsic geometry of convex ideal polyhedra in hyperbolic \(3\)-space. In: Gyllenberg, M., Persson, L.-E. (eds.) Analysis, Algebra, and Computers in Mathematical Research. Lecture Notes in Pure and Appl. Math., vol. 156, pp. 275–291. Dekker, New York (1994)

    Google Scholar 

  32. 32.

    Sakuma, M., Weeks, J.R.: The generalized tilt formula. Geom. Dedicata 55(2), 115–123 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Schlenker, J.-M.: Hyperbolic manifolds with polyhedral boundary. arXiv:math/0111136 (2001)

  34. 34.

    Schlenker, J.-M.: A rigidity criterion for non-convex polyhedra. Discrete Comput. Geom. 33(2), 207–221 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Springborn, B.A.: A variational principle for weighted Delaunay triangulations and hyperideal polyhedra. J. Differ. Geom. 78(2), 333–367 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: Rivin, I., Rourke, C., Series, C. (eds.) The Epstein Birthday Schrift. Geometry & Topology Monographs, vol. 1, pp. 511–549. Geometry & Topology Publications, Coventry (1998)

    Google Scholar 

  37. 37.

    Tillmann, S., Wong, S.: An algorithm for the Euclidean cell decomposition of a cusped strictly convex projective surface. J. Comput. Geom. 7(1), 237–255 (2016)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Weeks, J.R.: Convex hulls and isometries of cusped hyperbolic \(3\)-manifolds. Topology Appl. 52(2), 127–149 (1993)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

This research was supported by DFG SFB/Transregio 109 “Discretization in Geometry and Dynamics”.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Boris Springborn.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Editor in Charge: Kenneth Clarkson

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Springborn, B. Ideal Hyperbolic Polyhedra and Discrete Uniformization. Discrete Comput Geom 64, 63–108 (2020). https://doi.org/10.1007/s00454-019-00132-8

Download citation

Keywords

  • Decorated Teichmüller space
  • Penner coordinates
  • Horocycle
  • Discrete conformal equivalence
  • Triangulated surface

Mathematics Subject Classification

  • 57M50
  • 52B10
  • 52C26